Theano学习笔记（四）——导数

来源：互联网发布：淘宝商城入驻流程编辑：程序博客网时间：2024/05/22 00:05

导数使用T.grad计算。

这里使用pp()打印梯度的符号表达式。

第3行输出是打印了经过优化器简化的符号梯度表达式，与第1个输出相比确实简单多了。

fill((x** TensorConstant{2}), TensorConstant{1.0})指创建一个x**2大小的矩阵，并填充1。

importtheano.tensor as Tfromtheano import ppfromtheano import functionx= T.dscalar('x')y= x ** 2gy= T.grad(y, x)printpp(gy)f= function([x], gy)printf(4)printpp(f.maker.fgraph.outputs[0])>>> ((fill((x** TensorConstant{2}), TensorConstant{1.0}) * TensorConstant{2}) * (x **(TensorConstant{2} - TensorConstant{1})))8.0(TensorConstant{2.0}* x)

T.grad的第1个参数必须是标量

例如计算逻辑函数sigmoid的导数：

$\frac{{ds\left( x \right)}}{{dx}} = s\left( x \right) \cdot \left( {1 - s\left( x \right)} \right)$

importtheano.tensor as Tfromtheano import functionx= T.dmatrix('x')s= T.sum(1 / (1 + T.exp(-x)))gs= T.grad(s, x)dlogistic= function([x], gs)printdlogistic([[0, 1], [-1, -2]])>>> [[0.25        0.19661193] [ 0.19661193 0.10499359]]

计算雅克比（Jacobian）矩阵

雅克比矩阵是向量的一阶偏导数：

$J = \left[ {\begin{array}{*{20}{c}}{\frac{{\partial {u_1}}}{{\partial {x_1}}}}&{\frac{{\partial {u_1}}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {u_1}}}{{\partial {x_n}}}}\\{\frac{{\partial {u_2}}}{{\partial {x_1}}}}&{\frac{{\partial {u_2}}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {u_2}}}{{\partial {x_n}}}}\\ \vdots & \vdots & \ddots & \vdots \\{\frac{{\partial {u_n}}}{{\partial {x_1}}}}&{\frac{{\partial {u_n}}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {u_n}}}{{\partial {x_n}}}}\end{array}} \right]$

用T.arrange生成从0到y.shape[0]的序列。循环计算。

scan可以提高创建符号循环效率。

lambda~是python内建的magicfunction.

x= T.dvector('x')y = x ** 2J, updates = theano.scan(lambdai, y,x : T.grad(y[i], x), sequences=T.arange(y.shape[0]), non_sequences=[y,x])f = function([x], J,updates=updates)f([4, 4])>>> [[ 8.  0.] [ 0. 8.]]

计算海森（Hessian）矩阵

海森矩阵是多元函数的二阶偏导数方阵。

$H\left( f \right) = \left[ {\begin{array}{*{20}{c}}{\frac{{{\partial ^2}f}}{{\partial x_1^2}}}&{\frac{{{\partial ^2}f}}{{\partial {x_1}\partial {x_2}}}}& \cdots &{\frac{{{\partial ^2}f}}{{\partial {x_1}\partial {x_n}}}}\\{\frac{{{\partial ^2}f}}{{\partial {x_2}\partial {x_1}}}}&{\frac{{{\partial ^2}f}}{{\partial x_2^2}}}& \cdots &{\frac{{{\partial ^2}f}}{{\partial {x_2}\partial {x_n}}}}\\ \vdots & \vdots & \ddots & \vdots \\{\frac{{{\partial ^2}f}}{{\partial {x_n}\partial {x_1}}}}&{\frac{{{\partial ^2}f}}{{\partial {x_n}\partial {x_2}}}}& \cdots &{\frac{{{\partial ^2}f}}{{\partial x_n^2}}}\end{array}} \right]$

只要用T.grad(cost,x)替换雅克比矩阵的一些y即可。

x= T.dvector('x')y = x** 2cost= y.sum()gy =T.grad(cost, x)H,updates = theano.scan(lambda i, gy,x : T.grad(gy[i], x),sequences=T.arange(gy.shape[0]), non_sequences=[gy, x])f =function([x], H, updates=updates)f([4,4])>>> [[2.  0.] [ 0. 2.]]

雅克比右乘

x可以由向量扩展成矩阵。雅克比右乘使用Rop:

W = T.dmatrix('W')V =T.dmatrix('V')x =T.dvector('x')y =T.dot(x, W)JV =T.Rop(y, W, V)f =theano.function([W, V, x], JV)printf([[1, 1], [1, 1]], [[2, 2], [2, 2]], [0,1])>>> [2.  2.]

雅克比左乘

雅克比左乘使用Lop:

$\frac{{\partial f\left( x \right)}}{{\partial x}}v$

import theanoimport theano.tensor as Tfrom theano import functionx = T.dvector('x')v =T.dvector('v')x =T.dvector('x')y =T.dot(x, W)VJ =T.Lop(y, W, v)f =theano.function([v,x], VJ)print f([2, 2], [0, 1])>>> [[0.  0.] [ 2. 2.]]

海森矩阵乘以向量

可以使用Rop

$v\frac{{\partial f\left( x \right)}}{{\partial x}}$

import theanoimport theano.tensor as Tfrom theano import functionx= T.dvector('x')v= T.dvector('v')y= T.sum(x ** 2)gy= T.grad(y, x)Hv= T.Rop(gy, x, v)f= theano.function([x, v], Hv)printf([4, 4], [2, 2])>>> [4.  4.]

0 0